Surface Science 62(1977)267- 276 North-HollandPubhshlngCompany S UR F AC E F R E E E NE RGI E S OF SOLI DMETALS" ES TI MATI ONFROM LI QUI DS URF ACE TENS I ONME AS URE ME NT SWRTYS ON Ferrous Metals Section,Phystcal Metallurgy Research Laboratories, Canada Centre for Minerals and Energy Technology, DepartmentofEnergy, Mines and Resources, Ottawa, Canada and W. A. MI L L E R DepartmentofMetallurgy and Matertals Sctence,Unwerslty ofToronto,Toronto,Canada Recewed25May1976,manuscriptreceived mfinalform24September1976 Anequationisderivedonsemi-theoreticalgroundswhichexpressesthesohd- vapour sur- lacefreeenergyasafunctionof thehquldsurfacetensionandthesohd- hqul dlnterfaclal free energyAmeansof calculatingrehablevaluesforthes ohd- hqmdenergytspresented,which t henallowsanaccurateestimateof sohdsurfaceenergyatthemeltmg temperature,Tin,tobe madeforthelargenumberofelementsforwhichdependablehqmdsurfacetensiondataexist AmethodofesttmatmgsurfaceentropyIS presented,andhasbeenusedtocalculate theener- giestypicalof "average ",high-indexsurfacesattemperaturesrangingfrom0KtoT mItis felt thatthlspaperdescribesthemostaccuratemethodpresentlyavailableforthecalculationof thesurfaceenergyofsohdsIn theabsenceof directexperimentalmeasurement 1.I n t r o d u c t i o nThe s ur f acee ne r gyo f sol i dsISani mp o r t a n t phys i cal p a r a me t e r mc ont r ol l i nga w~der angeo f p h e n o me n a , suchast he st ressf or br i t t l e f r a c t ur e , t her at eo f sInt er- i ng, andt h e g r o wt h r at edur i ngpar t i cl ecoar s eni ng. F o r t l usr eas on, t her ehaveb e e nma n y a t t e mpt s , bot he x p e r i me n t a l andt he or e t i c a l , t oder i veval ueso f t he sur f ace f r eee ne r gy, s umma r i e s havebe e ngi venr ecent l yb y Bl akel y[1], LI nf or d[ 2] , and Mykur a [3]The avai l abl ee x p e r i me n t a l t e c h n i q u e s aredi f f i c ul t andus ual l yInac- cur at e, wi t h r esul t sf r omdi f f e r e nt wor ke r s o f t e n bei ngInver yp o o r a gr e e me nt [ 4] .Th e o r e t i c a l me t h o d s arest rnfl arl yc o mp l i c a t e d , a l t hought her eISs omeh o p e t hatr ecent advances wallp r o d u c e r el i abl ec a l c ul a t i ons b o t h f or s i mpl eme t a l s [5]and f or t r a ns i t i onme t a l s [ 6] .The e x p e r i me n t a l d e t e r mi n a t i o n o f a bs ol ut e val ueso f s ur f aceener gyf or a ny t y p e o f i nt e r f a c e isb o t h di f f i cul t ands ubj ect t on u me r o u s er r or s , t he mos t sever e 267 268W RTyson,W AMdler/Surfacefreeenergteso fsohdmetals ofwhichISusuallyloweringofthetruelnterfaclalenergybysurface-activecon- tarmnants.ThesemaybeIntroducedeitherasdissolvedImpurityfromwithinthe materialorasanadsorbingspeciesfromthesurroundinggaseousatmosphere Nevertheless,surfacefreeenergy(surfacetension)canbemeasuredmuchmore accuratelyforhqut ds thanforsolids(althoughnotnecessarilymoreeasily),and comprehensivetabulationsof hquld-vapoursurface tension 7LVareavailablefor a largenumberof elements[7].It is thepurposeof thispapertodescribebriefly how theliquidsurfacetension datamay beusedtoderivereliableestimatesof "average" sohd-vapoursurfaceenergies7sv, thebasisofthemethodwillbeanalyzedIn greater detailelsewhere[8] 2.Estimation of 7SV/TLV at themelting point,Tm Threemterfaclalfreeenergiesarerevolvedmsolld-hquld-vapourequlhbrium 7sv, 3'LV,andthesolld-lNuldlnterfaclalfreeenergy7SLInthissection,we attempttodeducetherelationshipsbetweenthesequantitiesmsinglecomponent systems Wemaywriteoneconditionlmme&atelywhichfollowsfromthewell-known observationthatmetalscannot normally besuperheatedabove T mThis Impliesthe absence of a nucleation barrierfor surface melting, hence ")'SV ~"~LV +')'SL( 1 )Notethatanytypeofhighenergydefectinthesohdsuchasdislocations,grain boundanes,surfacesteps,etc.,maybeabletonucleatemeltingwithoutanenergy barrierIn thepresentanalysishowever, weassumethata local region of a perfectly flat,Infinitesurfacemay alsonucleatemelting withnosuperheating,so thateq.(I) may beapplied. AttemperaturesclosetoTin,itwouldbeexpectedthatthesohdsurfacehas a structurewhichIS atomicallyroughIf thisroughening IS achievedwithoutdestroy- ingthecrystallographicnatureof thesurface, thenthesurfacecan beconsidered to be"crystalline"Iftherougheningandatomicrearrangementismoresevere,then thestateofthesurfacemaybemoreakintoaliquid,andasurface"melting" transitionmayoccurbelowTmThenatureof theprocesswhichactuallyoccur, willbedeterminedbythemagnitudes of thevarious freeenergy changes. Supposethatthesurface wereto remaincrystallineuptoT mThen crystallogra- phicrougheningwouldoccur,withanassociatedsurfacefreeenergy"yR. Suppose, ontheotherhand,thatsurfacemelting were tooccur attemperaturesbelowTmto adepthsufficientforthesurfacelayertobeconsideredas a bulk liquidwithinde- pendentsohd-hquldandhquld-vapourInterfacesThesurfacefreeenergy of such a solidwould be 7M =TLv+TsL +(GLGs) t ,W RTyson,W AMdler/Surface freeenergiesof s ohdmetals269 whereGL, G Sarethefreeenergiesperumtvolume of the hqmdandsohd,respec- tively,andtisthedepthof thelnterfaclalhqmd layer.At temperaturesbelowTm, thelasttermincreasesthefreeenergy andin order tominimize its magnitudetwill clearlybesmall,oftheorderofatmostafewmonolayerthicknesses.Hence,the lasttermtends tozeroatTm,as (G L -Gs) tends tozero Unfortunately,therearenoexpenmentaldatawithwhichtoestabhshrehably thestructureofsolidsurfaces attemperaturesclosetoT m.It is knownfromLEED studiesthatmetalsurfacesarecrystallmeattemperatureswellbelowTm,butitIs notknownwhetherthesurfacemeltsbeforeT misreached.Thestateofthe surface,andhence7sv, willbedeterminedbytherelativemagnitudesof ?'Rand 7M.Asshownabove,thesurfaceenergyof asolidcoveredbyathinmolten layer is "i'M =")'SL + ")'LV atTm, andso surface meltingshould takeplaceattemperatures equaltoandslightlylessthanTmunless7R~